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Expectation value of two annihilation operators

  1. May 16, 2009 #1

    I was studying about the effect of a beam splitter in a text on quantum optics. I understand that if a and b represent the mode operators for the two beams incident on the splitter, then the operator for one of the outgoing beams is the following,
    [tex] c = \frac{(a + ib)}{\sqrt{2}} [/tex]​

    Now if I try to measure the intensity of this beam by a photodiode, the intensity will be proportional to-

    [tex]<c^{\dag} c>[/tex]

    On evaluating this, I get,
    [tex]\frac{\left(<a^{\dag} a> + <b^{\dag} b> + i(<a^{\dag} b> - <b^{\dag} a>)\right)}{2} [/tex]
    Now the book says, that this can be written as,
    [tex]\frac{\left(<a^{\dag} a> + <b^{\dag} b> + i(<a^{\dag}><b> - <b^{\dag}><a>)\right)}{2} [/tex]

    I am unable to understand this step, that is [tex]<a^{\dag} b> = <a^{\dag}><b>[/tex]
    can someone please explain this.
    I understand that these mode operators commute, but is this always true for any two commuting operators.

  2. jcsd
  3. May 16, 2009 #2


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    if you write what [itex] < a^{\dagger} \, b > [/itex] is, then I am sure that you can figure it out.
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